Odifreddi's book successfully portrays the major developments in 20^{th} century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years. In the process, it traces the early historical context from which such problems have emerged, thereby linking the respective areas of contemporary mathematics to their classical origins. As such, it comes very near to being that intangible entity — a history of modern mathematics. Moreover, the literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification.

Speaking of literary style, one should bear in mind that the book was first published in Italian in 2000 and translated into English in 2004 by Arturo Sangalli. Without access to the original, it is difficult to pronounce with certainty as to the accuracy of the translation, except to say that, apart from two notable slips of between the tongues, the style is fluent and the mathematical narrative is generally sound.

The success of this book is partly due to the way in which its contents are structured but mainly to the way in which the problems are classified. There are just five chapters, as follows:

**Chapter 1: 'The Foundations':** Sets, Structures, Categories, Functions.
**Chapter 2: Pure Mathematics:** Analysis, Algebra, Number Theory, Topology, Gödel's Incompleteness Theorem, Model Theory, Discrete Geometry, etc.
**Chapter 3: Applied Mathematics:** Crystallography, Tensor Calculus, Game Theory, Functional Analysis, Optimisation Theory, Dynamical Systems, etc.
**Chapter 4: Mathematics and the Computer:** Theory of Algorithms, Artificial Intelligence, Chaos Theory, Computer Assisted Proofs, Fractals.
**Chapter 5: Open Problems:** Perfect Numbers, the Riemann Hypothesis, the Poincaré Conjecture, Complexity Theory.

Under these headings, the author provides much more detail than indicated above. For instance, with respect to algebra in chapter 2, there are sections allocated to Gorenstein's classification of finite groups and Steinitz's classification of fields. For topology, there are sections on Thurston's classification of 3d surfaces and Milnor's exotic structures and so on.

Discussion of the problems runs along two parallel tracks, the first of which is based upon thirteen of the Hilbert problems together with a historical survey of another twenty-one problems, beginning with Euclid's work on perfect numbers and culminating with the Moonshine Conjecture (amusingly mistranslated as the "Moonlight" Conjecture) of Conway and Norton (1979). Included this time span there is also mention of problems tackled by Kepler (configuration of spheres), Fermat, Goldbach, Riemann, Cantor, Mertens (bounds on the Möbius M function) and so on.

The second track is devoted to discussion of mathematical results that have led to the award of major prizes such as the Fields Medal, Wolf Prize, Nobel Prize and the Turing award and there is a total of sixty-six problems within these three categories.

The first chapter (Foundations) is by far the shortest, but it nicely sets the tone for the rest of the book. It illustrates how 'foundational' matters have been of importance since the time of the Pythagoreans and it covers issues like the *extensionality principle* of Leibniz and moves on to those well-known matters addressed by Frege, Russell and Cantor. This is followed by consideration of other foundational notions, such as the Bourbaki structural viewpoint, category theory and the lambda calculus of Alonzo Church. Odifreddi introduces the main players in these areas of work and he concludes with an explanation as to why Dana Scott earned the Turing Award in 1976.

Chapter 2 (Pure Mathematics) covers fifteen mathematical themes and mentions a myriad of mathematicians in the process. To convey the historical approach used by the author, a look at one of these topics may suffice. For example, section 2.11 explains the emergence of singularity theory and Thom's classification of catastrophes. Here, the starting point the work on conics by early Greek mathematicians, followed by mention of Descartes' use of algebra for the purpose of classification. Newton comes next by virtue of his work on cubics. We then read that, in 1740, J-P de Gua de Malves discovered that singular points of algebraic curves may be obtained by composing knots, cusps and inflexions in various ways. Proceeding to the 20^{th} century, the author reviews the work of Marston Morse and the theorem that characterises singular points. He introduces us to Hassler Witney and his work on cusps (Wolf prize 1982), which inspired René Thom to undertake his work on the classification of catastrophes. This section culminates with discourse on the related achievements of John Mather, Christopher Zeeman and Igor Prigogine (Nobel Prize 1977).

Another section epitomising the flavour of the text is 2.12 (Gorentstein's classification of finite groups). This is almost a mini history of group theory that begins with Bablyonian knowledge of solutions of quadratics goes on to discuss major achievements in the field of Lie groups and ends with mention of John Thompson's work on the second Burnside conjecture (Fields medal, 1970).

Chapter 3 is devoted to applied mathematics and its ten sections are concerned with:

- Crystallography: Bierbach's symmetry groups (1910)
- Tensor calculus: Einstein's general theory (1915)
- Game theory: Von Neuman's minimax theorem (1928)
- Functional analysis: Von Neuman's axiomatization of quantum mechanics (1932)
- Probability theory: Kolmogorov's axiomatization (1933)
- Opitmization theory: Dantzig's simplex method (1947)
- Equilibrium theory: Arrow-Debreu existence theorem (1954)
- Formal languages: Chomsky's classification (1957)
- Dynamical systems: The KAM theorem (1962)
- Knot theory: Jones invariants (1984)

Of the '30 greatest problems of the last 100 years,' Odifreddi has chosen the above as his ten candidates from the area of applied mathematics and the accompanying historical narrative maintains the same breadth and quality of the preceding chapters.

The penultimate chapter yields another five of the author's chosen 30, which, in order of appearance, represent the achievements of Turing, Shannon, Lorenz, Appel and Haken and Mandelbrot. The chapter heading is, of course, 'Mathematics and the Computer'.

Obviously, a book whose theme is mathematical problem solving would be incomplete without reference to unsolved problems. So, by way of conclusion, the final chapter is devoted to this very subject. Accordingly, Odifreddi places emphasis on four particular examples:

- The perfect numbers problem
- The Riemann hypothesis
- The Poincaré conjecture
- The P = NP conjecture (complexity theory)< /li >

However, the distinction between problems that are presently *unsolved* and problems that are deemed *unsolvable* is another matter.

So, what *are* Odifreddi's 30 greatest problems? He enumerates 15 from the areas of applied and computer-related mathematics (as above) and this review has mentioned several of those he has chosen from the realm of pure mathematics. As for the remainder, purchase of this excellent little book (little in size, massive in scope!) will reveal their identity.

Peter Ruane ([email protected]) has been involved with the teaching of mathematics at many levels since 1966. He has taught children from five to eighteen years old and he was employed in the field of mathematics education of teachers for over 25 years. Prior to his career in mathematics education he was variously employed, for a period of twelve years, as a docker (stevedore), slaughterman, bartender, farm labourer, hotel worker, shop assistant, etc. One of his ambitions is to be able to write books like Odifreddi's!